On the global generation of higher direct images of pluricanonical bundles

Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the effective generation of the higher direct images $R^i{f}_{\ast }(K_X^m)$, where $K_X^m$ is the $m$-th tensor power of the canonical bundle of $X$. In particular, we answer two questions posed by Popa–Schnell in [PS14].